University competition, grading standards, and grade inflation.

Popov, Sergey V. ; Bernhardt, Dan

I. INTRODUCTION

Universities award grades to measure the performance of students in courses. In turn, important decisions by third parties are based in part on grade point averages (GPAs)--firms tend to offer higher wages to students with high GPAs, and graduate schools tend to admit high GPA students. (1) In this article, we study how universities choose grading standards when they care about the decisions made by third parties based on GPAs. We then characterize how student body qualities at different schools interact with the depth of the job market to affect equilibrium grading standards.

We show how this strategic competition between universities can reconcile three central empirical regularities describing grading over the past 50 years: (1) GPAs are higher at better schools, (2)GPAs have risen over time at all schools, and (3)grading standards have fallen faster over time at better schools.

It is manifest that better universities award more high grades. For example, Rojstaczer (2003) finds that GPAs at private universities in 2006-2007 were 0.3 higher than at public universities. Figure 1 reinforces these findings, presenting the evolution of grades at selected universities between 1960 and 2000. This figure reveals that grades at better universities are uniformly higher. The figure also highlights a uniform secular rise in GPAs over time. In addition, over the entire sample period, GPAs at better universities increased significantly faster: the mean grade inflation at better universities significantly exceeds 0.5933 (p value .0073), the average increase at lesser universities, although there is no significant difference in grade inflation in different universities between 1980 and 2000 (Table 1).

We develop a model in which universities are distinguished by the distributions of "academic abilities" of their students: the distributions of student academic abilities at better schools conditionally stochastically dominate those at lesser schools. Firms value both academic ability and non-academic skills that firms can observe directly, which are complements in production. There are two types of jobs, good and bad, which are distinguished by the higher marginal product of skills in good jobs. Good jobs are in limited supply. Universities determine which students receive "A" grades by setting endogenously chosen cutoffs on academic ability. Firms learn some aspects of a student's non-academic skills via job interviews, and forecast academic abilities using the information contained in the ability distribution at a student's university, the university's grading standard, and the student's grade. Firms then assign jobs: students with higher expected productivities are assigned "good" jobs where the marginal product of skill is higher.

[FIGURE 1 OMITTED]

Universities understand how firms determine job placement and wages, and set grading standards to maximize the total expected wages of their graduates. Top universities would argue that their higher proportions of high grades simply reflect their better student bodies; a common grading standard would inevitably lead to more good grades at better schools. A central finding of our analysis is that under weak conditions, top universities set softer grading standards: the marginal "A" student at a top university is less able than the marginal "A" student at a lesser university. The intuition for this result devolves from the basic observation that a marginal student at a top school can free ride on the better upper tail of students because firms cannot distinguish "good A" students from "bad A" students. In contrast, lesser schools must compete for better job assignments by raising the average ability of students who receive "A" grades, setting excessively high grading standards.

Importantly, a social planner who seeks to maximize total output in society would choose the opposite ordering, setting more demanding grading standards at top schools whenever there is heterogeneity among students in observable non-academic skills. If academic skill was the only source of heterogeneity, then the social planner would set the same grading standard at each school. However, then the expected academic productivity of students with "A"s at top schools is higher. Firms, which do not see the abilities of students, would then "overrate" marginal "A" students from top schools, assigning too many with low observable, nonacademic (hereafter referred to as "social") skills to good jobs. The social planner wants to equalize the expected productivity of the marginal "A" students who receive good job assignments, and since the average social skill is less for students from top schools with good job assignments, the social planner sets a higher grading standard for "A"s at top schools.

Finally, we suggest a plausible driving force underlying grade inflation at universities: the secular increase over time in the measure of good jobs relative to the measure of students, possibly reflecting the well-established shift toward skill-biased technologies. (2) There is extensive evidence that skill demands at jobs have increased significantly, suggesting that there are now far more good jobs. We show that universities respond to an increase in good jobs by trying to place less able students at good jobs. In particular, universities reduce cutoffs for "A" grades.

The upward trend in grades is perhaps less interesting than the inference problems it creates for third parties. In and of itself, grade inflation might not increase inefficiency in the economy. However, we provide conditions under which grade inflation exacerbates differences in grading standards, further distorting hiring decisions. That is, we provide conditions under which the cutoff for an "A" falls by more at top schools. Intuitively, if there are very few good jobs, grade standards must be very high, so that with a common bounded support on ability, the ability differences between the marginal "A" and average "A" student must be small at all schools. However, when more students get "A"s, this difference grows, and top schools exploit this via lowering their grading standards by more. We provide conditions under which this greater reduction in grading standards at top schools is associated with a greater increase in the number of "A"s, that is, for grade inflation at top schools to be higher.

We next review the literature. Section II presents the basic model of university competition. Section III characterizes equilibrium and social planner outcomes when all students have the same social skill. Section IV considers students with heterogeneous social skills. Section V summarizes our findings and discusses their robustness. The Appendix contains all the proofs.

A. The Literature

Yang and Yip (2003) develop a free-rider paper that also predicts that better schools should give more high grades. Their mechanism is different: In their model, universities destroy value by explicitly lying about the ability of a student to perform a good job, with the result that in their interior equilibrium, the expected productivity of high-grade students leaves firms completely indifferent between assigning high-grade students to good and bad jobs. Our framework also allows for heterogeneity across student types, along both observable and unobservable abilities. Boleslavsky and Cotton (2012) consider a setting similar to ours with two student abilities, endogenizing investment in school quality. They show that without strict adherence to complete revelation in grading, schools might give less able students high grades in equilibrium; but, contrary to conventional wisdom, such upward distortion in grades can increase investment in school quality, as the environment becomes more competitive.

Chan, Hao, and Suen (2007) consider a setting in which the measure of good students at a school is random, observed by schools, but not by firms. They model the intentional loss of academic reliability where the grading standard is not fixed, so that otherwise identical students who take identical actions might receive different grades. They argue that this is why grading standards have varied over time. However, they are silent as to why there should be significant unobservable year-to-year variation in the quality of large populations of students at a university, especially given indirect, but broadly observable measures of student quality such as mean SAT (Scholastic Assessment Test) and ACT (American College Testing) scores, and measures of high-school class rank.

Dubey and Geanakoplos (2010) investigate how discreteness of grades influences student effort: they find that when students only care about relative rank, coarser grade structures can motivate students to study harder. Without noise in assessment ability, there is little reason to have a big support for grades, at least at the right tail. Ostrovsky and Schwarz (2010) study the optimal transcript structure problem, and find reasons to give the same grade to a sizeable fraction of students in the right tail, when many schools have worse ability distributions. Zubrickas (2010) studies the optimal mechanism for eliciting effort from students by awarding grades, and finds that with imperfect ability assessment, to elicit effort from the best students there must be a positive mass of students with the best grade. MacLeod and Urquiola (2009) explore how the structure of the schooling market affects tradeoffs between studying effort, wealth, and leisure.

A body of literature studies grading standards from a central planner's perspective. Costrell 1994 studies how different policies toward standards affect student effort (he states that an egalitarian central planner is likely to set lower standards than a total earnings maximizer), and he reviews the grading literature; Betts (1998) makes an opposing argument.

II. THE COMPETITION BETWEEN UNIVERSITIES

The world contains universities, distinguished by type u [member of] U, where U [subset] R is compact. University types are distinguished by the ability distributions of their student bodies. We use H and I to index representative examples of universities with "better" and "worse" distributions of student academic abilities. Student academic abilities at a type u school are distributed according to a continuous and strictly positive density [f.sub.u](x) on a support, [[[theta].bar], [bar.[theta]]], that is common to all universities. We capture the notion of a "better" student body with the concept of conditional first-order stochastic dominance (3): [f.sub.H]([theta]|[theta] > t) first order stochastically dominates [f.sub.I]([theta]|[theta] > t) for all t [member of] [bar.[theta]]), written [f.sub.H] [??] c [f.sub.I]. In particular, the associated cumulative distribution functions satisfy [F.sub.I]([theta]|[theta] > t) > [F.sub.H]([theta]|[theta] > t), for all t [member of] ([[theta].bar], [bar.[theta]]) and [theta] [member of] (t, [bar.[theta]]). Without loss of generality, we assume that [f.sub.H] [??] c [f.sub.I] if and only if H > 1, for H, I [member of] U.

To capture the fact that any single university admits a negligible portion of the entire pool of students, we assume that each university has a measure zero of students. We normalize the total measure of students to one. We use [alpha](u) to represent the distribution over university types. Thus, [[integral].sub.u[member of]U] [F.sub.u] (q) d[alpha] (u) is the measure of students with academic abilities below q.

A student is distinguished by three attributes: (1) his university type u, (2) his academic ability, [theta], and (3) a non-academic skill component [mu] that firms can learn via job interviews, but does not affect academic performance. We refer to this non-academic skill as a "social" skill. We assume that social skills, IX, are distributed according to a strictly positive density g(x) and distribution G(x) with non-negative full interval support [[[mu].bar], [bar.[muy]]], independently of academic skills. We initially assume that the distribution of social skills is the same at all schools. This assumption is consistent with the observation that universities largely filter students via high-school academic performance and various academic tests such as the SAT or ACT or essays. We make the standard increasing hazard rate assumption on g and [f.sub.u]. We address the possibility that better schools may have students who tend to have both better academic and nonacademic social skills.

There are two types of jobs. There is a positive measure [GAMMA] of "good" jobs where the productivity of a student with ability ([theta], [mu]) would be S[theta][mu], and many "bad" jobs in which the student's productivity would be s[theta][mu], where S > s > 0. As in Yang and Yip (2003) and Coate and Loury (1993), we assume that everyone at the same job receives the same wage. Students with good job placements receive wage W and those placed at bad jobs receive wage w, with W > w > 0. (4) Via job interviews, firms can observe [mu], but they do not directly observe [theta]. Profit maximization immediately implies that firms would like to hire more productive students, and complementaries between ability and job quality ensure that firms place students whom they expect to be sufficiently productive in good jobs. Universities know the academic abilities of their students, but not their social skills. (5) Their problem is to assign a grade g [member of] {A, B} to each student. Universities choose grading standards to maximize the expected sum of wages earned by graduates. One can equivalently view job placement in good and bad jobs as placement in good and bad graduate or professional programs, where universities want to place more students in good programs. In conclusion, we discuss how one should interpret our analysis when students take multiple courses, and receive both A and B grades.

Firms do not observe student academic abilities. However, firms know the alma mater of each student and the distribution of academic abilities at each school, and hence can extract information about academic abilities from grades.

A. Decisions of Agents

We assume that universities adopt grading strategies that take the form of a cutoff, so university u gives a student an "A" if and only if his academic ability [theta] exceeds a cutoff [[??].sub.u] chosen by the university. The same equilibrium outcome would emerge were the labels "A" and "B" reversed: we adopt the convention that an "A" grade refers to the better subpopulation of students. In addition, since giving all students "A" grades is equivalent to giving all students "B" grades, without loss of generality, we assume that if it is optimal for a university to give all students the same grade, then it gives all students "A" grades, as at Doonesbury's fictional Walden University.

More generally, our model is sufficiently sparse that equilibria can exist in which grading strategies do not take a cutoff form. Non-cutoff strategies can emerge in equilibrium simply because once firms form beliefs (which determine job assignments), they are not affected by which set of student abilities receive "A"s. However, such equilibria are not robust to natural refinements that pin down what universities do. For example, non-cutoff strategies cannot be part of an equilibrium if (1) firms observe the true ability of a small measure of students (and schools do not know which ones), from which firms infer average productivities of "A" and "B" students, or (2) universities care about which students are placed where, so that, for example, they prefer that better students are placed in better graduate programs for reputational reasons. Under such scenarios, universities have strict incentives to ensure their most able students receive "A"s, in which case equilibrium grading strategies take cutoff forms.

We denote a student from a type u university with grade g as a ug student. The expected academic ability of a ug student is [E.sub.ug][[theta]] = ([[integral].sup.[bar.[theta]].sub.[[theta].bar]] I (grade is g) [theta] [f.sub.u]([theta])d[theta]) / ([[integral].sup.[bar.[theta]].sub.[[theta].bar]] I(grade is g) [f.sub.u] ([theta])d[theta]). (6) Notice that issuing fewer "A's raises the expected academic ability of both "A" and "B" students: increasing the grading standard [[??].sub.u] increases both [E.sub.uA][[theta]] and [E.sub.uB][[theta]]. Also, if the universities were to set a common grading standard, then the better type H university would have more "A" students than type I because [f.sub.H]([theta]) [??] c [f.sub.I] ([theta]).

B. Labor Market Outcomes

Students prefer good jobs placements to bad ones, as W > w. Since firms profit from placing more able students in good jobs, there will be an endogenous threshold K on a student's expected productivity such that firms will place students in good jobs if and only if their expected productivity is at least K. We use this productivity threshold to determine the university-specific threshold [[??].sub.ug] = K/([E.sub.ug][[theta]]) on social skills, such that, at university u, only students with grade g and social skills exceeding this standard receive good jobs. We assume that it is profitable in equilibrium to fill both good job and bad job vacancies.

C. University Decision

Given an equilibrium productivity standard K [less than or equal to] [bar.[theta]] [bar.[mu]] for a good job, a type u university maximizes the total income of its student body by choosing the [[??].sub.u] that solves:

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Subtracting the total wage of all students of university u were they all employed on bad jobs (a constant that does not depend on the grading standard) from [[pi].sub.u], we can rewrite the university's objective as

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Dividing this result by (W - w) > 0 yields:

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Thus, a university maximizes the total wage bill of its student body by maximizing the total employment of its students at good jobs. (7)

An immediate implication is that some "A" students from each university always receive good jobs. A student only receives a good job if his expected productivity exceeds the endogenous level K (associated with measure [GAMMA] of students receiving good jobs). The common support assumptions on academic abilities and social skills ensure that all universities have at least a few very able students. By setting a sufficiently high grading standard, a university can ensure that its "A" students have unobservable productivities arbitrarily close to [bar.[theta]]. Some of these students also have high social skills, and hence receive good jobs.

D. Equilibrium

A symmetric pure strategy equilibrium is a collection of grading standards {[[??].sup.*.sub.u]}, social skill cutoffs {[[??].sup.*.sub.ug]}, u [member of] U, g [member of] {A, B}, and a minimal productivity standard [K.sup.*] such that:

* [[??].sup.*.sub.u] maximizes the total alumni wage bill at a type u [member of] U university given productivity standard [K.sup.*];

* Firms' job assignments maximize profit given [K.sup.*]:

--[[mu].bar][E.sub.ug][[theta]|[[theta].sup.*.sub.u]] [greater than or equal to] if [[??].sup.*.sub.ug] = [[mu].bar] and all ug students receive good jobs;

--[bar.[mu]][E.sub.ug][[theta] | [[theta].sup.*.sub.u]] [less than or equal to] [K.sup.*], if [[??].sup.*.sub.urg] = [bar.[mu]] and no ug students receive good jobs;

--[[??].sup.*.sub.ug] [E.sub.ug][[theta]|[[theta].sup.*.sub.u]] = [K.sup.*], otherwise.

* [K.sup.*] "clears" the market: given [[??].sup.*.sub.u] and [[??].sup.*.sub.ug], the measure of students with expected productivity of at least [K.sup.*] is [GAMMA].

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The next proposition establishes the existence of a symmetric equilibrium (i.e., each type u school sets the same grading standard).

PROPOSITION 1. A pure strategy symmetric equilibrium exists. There is a unique equilibrium productivity standard [K.sup.*] for a good job, and the expected product of students from school type u who receive good jobs in equilibrium is unique.

Uniqueness of an optimal grading standard at each type u university is not guaranteed absent assumptions on the measure of good jobs. For example, were there so many good jobs that each student at a very good type u university receives one, then that university could achieve this by giving all students "A"s, or by giving only a few students "A"s, so that the expected productivities of its "B" students were high enough that they all receive good jobs.

III. NO HETEROGENEITY IN SOCIAL SKILLS

We begin by analyzing the special case in which all students have the same social skill level, which we normalize to [mu] = 1.

To solve for the equilibrium, first notice that it is never an equilibrium for some, but not all, students with grade g from university u to get good jobs. Were this so, a university could marginally increase its grading standard (if it cannot, it means that everyone is getting an "A," which is equivalent to giving everyone "B," which has plenty of room for marginally increasing grading standards), raising the average ability of both its "A" and "B" students. In turn, the expected productivity of all students with grade g is raised, so that all now receive good jobs.

Moreover, when [GAMMA] is sufficiently small, then only "A" students receive good jobs. One can derive an explicit upper bound on [GAMMA] that ensures that only "A" students receive good jobs. To do this, one can solve for the quantity of "A" students when the minimal required productivity is [E.sub.u] [[theta]|u = max U]--a necessary condition for the best university type to place all of its students at good jobs; if [GAMMA] is smaller than this upper bound, then awarding every student at a school an "A" is not part of an equilibrium, and only "A" students receive good jobs. Most transparently, when there are two types of schools H and I, with measure [alpha] of type H universities, then [GAMMA] < [alpha] is grossly sufficient to ensure that only "A" students receive good jobs.

So consider an equilibrium in which only "A" students receive good jobs. Were, say, HA students to have higher expected academic productivities than IA students, then a type H university could lower its standard for an "A" slightly, and more of its students would then receive good jobs. Therefore, in equilibrium, the expected productivity of "A" students at the two types of schools must be equal. Combined with the capacity constraint, this implies that equilibrium is fully described by:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The next result establishes that under a weak sufficient condition, better universities set slacker grading standards than dominated universities.

PROPOSITION 2. Suppose [GAMMA] is small enough that, in equilibrium, not all students at a university receive good jobs. Then, in equilibrium, [[??].sup.*.sub.H] < [[??].sup.*.sub.1] if and only if H > I.

Proposition 2 says that some students at type H universities receive "A"s, but their abilities are low enough that they would receive "B"s at a lesser type I university. Better universities have incentives to dilute the mass of good "A" students with students who would receive "B"s elsewhere: even after dilution, the better upper tail of good students at type H universities makes an average "A" student at a type H university as good as a typical "A" student from a lesser type I university.

A. Social Planner

A social planner can set and enforce grading standards at schools (say, by limiting the quantity of "A" grades), but cannot assign particular students to particular jobs. She seeks to maximize total output in the economy (which is equivalent to maximizing output on good jobs), taking into account how firms will assign good jobs given the grading standards that she sets. The social planner sets her grading standards to solve

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROPOSITION 3. The social planner sets a common grading standard, [[??].sup.P.sub.u] = [[??].sup.P].

The social planner ensures that the marginal "A" student at each university has exactly the same academic ability. This immediately implies that the expected productivity of "A" students is higher at better universities. However, when students all have the same social skills, this does not create any inefficiencies in placement: the "right" students receive good job assignments. This will not hold when students differ in social skills.

COROLLARY 1. Suppose [GAMMA] is small enough that some students from each university are placed at bad jobs. Then [[??].sup.*.sub.max U] < [[??].sup.p] < [[??].sup.*.sub.min U].

The corollary says that from a social perspective, sufficiently bad universities set excessively high grading standards, while sufficiently good schools set standards for "A" grades that are too low. The result follows since the good jobs capacity is exhausted in both the social optimum and equilibrium, so both [[??].sup.*.sub.max u] < [[??].sup.*.sub.min U] [less than or equal to] [[??].sup.P] and [[??].sup.P] [less than or equal to] [[??].sup.*.sub.max u] < [[??].sup.*.sub.min U] cannot occur.

Figure 2 conveys the intuition, depicting equilibrium and social planner outcomes when there are two types of schools. CFOSD implies that the solid line of equal expected productivities lies below the 45[degrees] line associated with equal grading standards; Proposition 2 says that the intersection of the dashed capacity constraint line and the solid equal productivity line is below the equal grading standards line; and the capacity constraint must be negatively sloped. Strategic considerations not only induce better universities to set slacker grading standards than is socially optimal, but they also force lesser universities to set stricter standards in order to compete. Still, type I universities prefer this outcome than the one in which no grades are disclosed, in which case no I alumni would receive good jobs.

[FIGURE 2 OMITTED]

We next show how the primitives of our economy affect outcomes when there are two types of schools, good (type H) and bad (type I), where the proportion of type H schools is [alpha]. Our results generalize to many types.

B. More Type H Universities

An increase in the fraction [alpha] of top schools flattens the dashed capacity constraint line because changes in grading standards at type H universities now have bigger impacts on the measure of students with As. There are conflicting effects: increasing [alpha] shifts the composition of schools from those with high standards to those with low standards; but increasing [alpha] increases the supply of able students. To determine whether grading standards improve, observe that a marginal increase in [alpha] is a (not necessarily shape preserving) counterclockwise rotation of the capacity line, and one point of the line remains the same. Whenever this point is above the 45[degrees] line, the intersection with the (blue) equal productivity line shifts up and grading standards rise. Moreover, when [alpha] = 1/2, the rotation point is above 45[degrees] line, since 1 - [F.sub.H](t) > 1 - [F.sub.I](t). Thus, there exists an [bar.[alpha]] < 1/2 such that for el > [bar.[alpha]], increasing the fraction of type H schools raises grading standards. That is, the supply effect dominates the composition effect when there are enough top universities.

C. Improvements in Student Body Composition

A first-order stochastic dominance improvement in the distribution of student abilities at a type 1 university causes type H universities to set higher standards for an "A," but has ambiguous effects on I's grading standard. To see this, notice that the "equal expected productivity" line shifts toward the 45[degrees] line (productivity of I university improves), and the capacity line moves to the right (for the same quantity of "A" students at a H university, one now needs a higher grading standard in I to fill the fewer remaining good jobs). As a result, the equilibrium [[??].sup.*.sub.H] must rise, but the effect on [[??].sup.*.sub.I] is ambiguous--the number of I students who receive good jobs must rise, but whether [[??].sup.*.sub.I] increases depends on whether or not the increased quality composition of good I students dominates the effect of having more good I students.

Intuitively, improving the distribution at type I schools creates added "competition" for type H schools forcing them to raise standards. The ambiguous impact on type I schools reflects that (a) type I schools can lower grading standards and still have a higher average quality of "A" students, but (b) the better distribution also increases competition for type I schools, raising the average ability required for a good job placement. Analogously, one can show that a deterioration in the distribution of abilities at type H universities eases grading standards at type I universities, but has ambiguous effects at type H schools.

D. Good Jobs, Grading Standards, and Grade Inflation

We next describe how increases in the number of good jobs affect equilibrium outcomes. It follows directly that increasing [GAMMA] causes all universities to lower their grading standards: this reduction in the average quality of "A" students implies that there is grade inflation. In particular, increasing [GAMMA] shifts the capacity line outward away from the top right corner, shifting equilibrium outcomes away along the "equal expectations about 'A' students" line. The social planner's choice shifts away along the 45[degrees] line, resulting in an equal decrease in grading standards and an increase in "A" grades at both universities.

We are especially interested in identifying when equilibrium grading standards fall by more at top schools, and when this translates into higher grade inflation at top schools. We take two types of schools, H and I, with [f.sub.H] [??] c [f.sub.I]. We characterize the relative impact of F on grading standards via the implicit function [[theta].sub.H]([[??].sub.1]), defined by [E.sub.H]([theta]|[theta] > [[theta].sub.H]) = [E.sub.I] ([theta]|[theta] > [[??].sub.I]) = K, by varying K (K falls with F): grading standards fall faster at type H schools than type I schools if and only if [[??]'.sub.H] ([[??].sub.I]) > 1.

PROPOSITION 4. Suppose there are sufficiently few good jobs, [GAMMA]. Then a slight increase in [GAMMA] causes grading standards to fall faster at type H schools than type I schools.

One would like to extend this result to settings where the number of good jobs is larger, maintaining only the premise that [GAMMA] is not so large that all H students receive good jobs. To do this, we consider a family of ability densities with linear right tails, where the linear right tail is "long" enough that it describes the abilities of A students:

(6) [f.sub.u]([theta]|[a.sub.u], [b.sub.u]) = [a.sub.u] + [b.sub.u][theta], [theta] [member of] [t, 1] for some t,

where, to ease presentation, we assume a [0, 1] support (extensions to a support [[theta].bar], [bar.[theta]]] are routine). Positivity of the density implies that [a.sub.u] + [b.sub.u] > 0 and [a.sub.u] + [b.sub.u]t > 0, and [f.sub.H] [??] c [f.sub.I] implies that [a.sub.H][b.sub.I] < [a.subl][b.sub.H]. We also need that [b.sub.H] sufficiently exceeds [b.sub.I]. (8) When densities are linear on their full support, [b.sub.H] > [b.sub.I] suffices.

PROPOSITION 5. Consider densities with linear right tails, where [b.sub.H] sufficiently exceeds [b.sub.I], so that [a.sub.H][b.sub.I](1 + [[??].sup.*.sub.I]) < [a.sub.I][b.sub.H](1 + [[??].sup.*.sub.H]). When [[??].sup.*.sub.u] > t, an increase in [GAMMA] causes grading standards to fall faster at type H universities than type I universities.

COROLLARY 2. If [f.sub.H]([[??].sub.H]) [greater than or equal to] [f.sub.I] ([[??].sub.I]) and grading standards fall faster in H, then the number of "A "s increases faster at type H universities than type I universities.

For example, if [f.sub.H] (t)/[f.sub.I] (t) increases in t, then there exists a [??] such that [f.sub.H](t') > [f.sub.I](t'), [for all]t' > [??], in which case Corollary 2 follows if and only if there are sufficiently few good jobs, [GAMMA].

In summary, when students only differ in academic ability, universities with better student bodies press some students from lesser universities out of good job assignments by setting slacker grading standards than is socially optimal. Further, under plausible scenarios, more good jobs give rise to greater grade inflation at top universities. We now explore how heterogeneity in social skills affects these conclusions.

IV. HETEROGENEOUS SOCIAL SKILLS

Suppose now that students differ in their non-academic social skills that firms observe. As a result, firms base job placement decisions on both the information about academic ability revealed by a student's university and grade, and the information about social skills that firms glean directly from job interviews. For heterogeneity in social skills to alter outcomes, there must be enough dispersion in social skills that firms do not assign good jobs to all "A" students. If not, then our previous analysis characterizes outcomes. We maintain the assumption that the dispersion is not sufficient for "B" students to receive good job assignments in equilibrium. Thus, the support [[[mu].bar], [bar.[mu]]] of the distribution of social skills G(x) is neither very small, nor very large; that is, heterogeneity in social skills is "intermediate" so that

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We now derive the implications of observable skill heterogeneity on equilibrium grading standards.

PROPOSITION 6. Suppose [[mu].sup.2]g([mu]) increases in Ix and heterogeneity in social skills is intermediate. Consider arbitrary university types H and I with [f.sub.H] >[sub.C] [f.sub.I]. Then, in the unique equilibrium, type I schools set higher grading standards than type H schools. Further, [[??].sup.*.sub.H] [less than or equal to] [[??].sup.*.sub.I] , where the inequality is strict as long as some A students do not receive good jobs.

The result says that as long as the density over social skills does not fall quickly, g'([mu]) > -2g([mu])/[mu], then top universities set slacker grading standards. When [GAMMA] is small enough that not all students at a school receive "A"s in equilibrium, and the dispersion in social skills is high enough that an "A" student with the lowest social skill is not offered a good job, then equilibrium is characterized by an interior solution, and a university's best response is characterized by the intersection of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The bottom equation, which characterizes how social skills affect firm placement of students with "A" grades, is decreasing in ([??], [??]) space (see Figure 3). Firms give good jobs to "A" students from university u with social skills [mu] [greater than or equal to] [[??].sub.[mu]], and bad jobs to all other students. The CFOSD assumption implies that [E.sub.I] [[theta]|[theta] > [??]] < [E.sub.H][[theta]]0 > [??]], for every [??]. Therefore, [[??].sup.*.sub.HA] < [[??].sup.*.sub.IA]. With our maintained assumption that [[mu].sup.2]g([mu]) increases in [mu], the right-hand side of the top equation increases in [mu]. This, combined with [[??].sup.*.sub.HA] < [[??].sup.*.sub.IA], implies [[??].sup.*.sub.H] < [[??].sup.*.sub.H].

COROLLARY 3. In any interior equilibrium, E[[theta][H, [theta] > [[??].sup.*.sub.H]] > E[[theta]|I, [theta] > [[??].sup.*.sub.I]].

The corollary follows directly from the optimal job assignment, [[??].sup.*.sub.HA]E[[theta]|H, [theta] > [[??].sup.*.sub.H]] = [[??].sup.*.sub.IA]E[[theta]|H, [theta] > [[??].sup.*.sub.I]], and Proposition 6, which states that [[??].sup.*.sub.HA] < [[??].sup.*.sub.IA]. Corollary 3 says that the average academic ability of an "A" student from a better university is higher than that of an "A" student from a worse school. In turn, firms set lower standards on social skills for students from better schools.

We now derive the qualitative impact of intermediate levels of heterogeneity in social skills on a social planner's grading standards.

PROPOSITION 7. Suppose heterogeneity in student social skills is intermediate, so that not all "A" students receive good jobs, and no "B" students receive good jobs. Then for two arbitrary schools of types H and I, with [f.sub.H] > [sub.C] [f.sub.I], the social planner sets [[??].sup.P.sub.H] > [[??].sup.P.sub.I] and [[??].sup.P.sub.HA] < [[??].sup.P.sub.IA]

The social planner sets grading standards so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The first equality confirms that a social planner sets grading standards in such a way that given the job assignment choices by firms, the expected productivity of the students employed on good jobs with the marginal social skills [[??].sup.P.sub.uA], a is equated at all schools. As a result, even though the distributions of social skills at universities are the same, the average social skill of "A" students from a type H university who receive good job assignments is less than the average social skill of "A" students at lesser schools. The second equality says that the social planner sets grading standards to equate the expected productivities of the marginal "A" student at each university along the [theta] dimension as well. Then, because the average social skill of students from a type H university with good jobs is less, it follows directly that a social planner sets more demanding grading standards at better universities.

[FIGURE 3 OMITTED]

The contrast with equilibrium outcomes is sharp: Proposition 6 reveals that, in equilibrium, better universities set slacker grading standards. In essence, the social planner wants to unwind the negative consequences of the labor market having imperfect assessments of academic abilities of students at different schools, and hence set higher standards at better schools. In contrast, in equilibrium, better schools want to exploit those imperfect assessments, and hence set lower standards.

One might conjecture that the different student body compositions at different universities hinder a social planner because it causes firms to distort hiring decisions toward students from better universities. This presumption is false: having different student bodies helps a social planner better condition good job assignments based on combinations of [mu] and [theta]. By setting higher standards at better schools, the social planner improves selection on total productivity of students placed at good jobs by firms. Another way to see how different university types benefit the social planner is to observe that firms set the same cutoffs on social skills as the social planner would. With heterogeneous universities, the social planner could set common grading cutoffs, but chooses not to. It follows that in an idealized world in which a social planner sets grading standards, homogenizing the university pool is always suboptimal.

In summary, as long as there is not so much heterogeneity in social skills that some "B" students receive good jobs, then under mild conditions on the density g(*) of observable nonacademic, social skills, better universities set lower grading standards, even though a social planner would make the opposite choice. These findings extend when some "B" students with exceptional skills also receive good jobs, as long as the distribution of social skills is such that "A" students dominate decision making. We believe that this is the relevant real-world scenario.

A. Heterogeneity in Non-Academic Skill Distributions

To ease exposition, we assumed that the distribution of non-academic skills in students that firms observe is the same at all universities. This +assumption allows us to compare how grading standards are set when universities differ solely on the dimension captured by their grading policies. We now show how our qualitative findings are reinforced when students at better universities tend not only to have better academic skills, but also better non-academic skills. We adjust the definition of equilibrium accordingly, but omit it for brevity. To ease analysis, we assume that [[mu].sup.2]g([mu]) is increasing, which is a weak sufficient condition for equilibrium to be characterized by the unique solution to the first-order conditions.

PROPOSITION 8. Let u and u' be two types of universities that have identical distributions of academic skills [F.sub.u](x) = [F.sub.u'], (x) but the students at university type u tend to have better observable social skills, [G.sub.u](x) > [sub.C] [G.sub.u'](x). Then the type u university sets a lower equilibrium grading standard than the type u' university, so that its social skill cutoff is higher.

A corollary of Propositions 6 and 8 is that if the universities with better populations of students with academic skills also have students with stochastically better observable non-academic skills, then this observable skill heterogeneity between universities serves to widen the gap between grading standards at better and worse schools. Consider two types of universities, H and I, where [F.sub.H] > [sub.C] [F.sub.I] and [G.sub.H] > [sub.C] [G.sub.I] (Figure 4). Consider a fictitious type H', which features the academic ability distribution [F.sub.H] and non-academic ability distribution [G.sub.I]. By Proposition 6, [[theta].sup.*.sub.H'] < [[theta].sup.*.sub.I]; by Proposition 8, [[theta].sup.*.sub.H] < [[theta].sup.*.sub.H']; and together the propositions imply that [[theta].sup.*.sub.H] < [[theta].sup.*.sub.I].

V. DISCUSSION AND CONCLUSION

The central message of our paper is that competition for good job assignments for graduates causes better universities to set lower standards for "A" s, because their marginal "A" students can ride on the coat tails of the better average qualities of "A" students. We show that a social planner sets the opposite ordering on grading standards. We also show that increases in the number of good jobs drives down standards for "A"s, and that under plausible scenarios, standards fall more at better schools.

[FIGURE 4 OMITTED]

This central message is robust to perturbing assumptions in many ways. For example, our analysis does not hinge on our common support assumption, as long as some students at a lesser schools get good jobs. So, too, one can show that the findings extend when (a) universities maximize placement of students in good graduate programs, or (b) workers receive wages equal to their expected productivities at their job given the information observable to firms including university and grades, or (c) wages are determined by bargaining in search settings, where students who receive good jobs bargain with firms to split the surplus associated with complementarities between skills and job quality.

The more important feature to contemplate is how to relate our simplified two-grade framework to the real-world setting in which students take many courses, and receive many grades over their four years. Analytical tractability precludes the analysis of such a setting. More importantly, approximating the real world with finely determined grades would be misleading because our model is a simplified representation of reality along other dimensions, as well. In reality, the cumulative GPAs that students receive are not noiseless, finely honed, 4-year assessments of ability. In fact, there is substantial residual uncertainty in cumulative GPAs about student abilities introduced by random components inherent to university education. For example, professors are imperfect ability evaluators; there is a random component to individual grades reflecting randomness inherent in exam construction, studying choices and luck; there is mis-match of students with professors or with courses; different students study very different fields and subjects; ability is a multi--dimensional attribute (e.g., analytic vs. language vs. writing abilities), weighed differently in different courses and fields of study; and so on. Moreover, with high-grade compression at the top end--with many students receiving A's--noise may determine most of the variation in the cumulative GPAs of better students.

The question that one must ask is: would the qualitative content of our findings extend? The central feature that we seem to need for the relevance of our analysis in this richer real-world setting is that firms weigh the "quality" of a student body distribution in their assessment of the content of individual student cumulative GPAs. In turn, this provides incentives for better universities to set grading standards that lead them to give "A" grades to more marginal students: such marginal students cannot be perfectly distinguished from the many superior students in their school who receive similar grades, and the presence of these superior students is weighted by firms in their assessments of individual students. The many anecdotal stories of better private universities catering to students suggest that this scenario is the relevant one.

ABBREVIATIONS

ACT: American College Testing

GPAs: Grade Point Averages

SAT: Scholastic Assessment Test

doi: 10.1111/j.1465-7295.2012.00491.x

APPENDIX

Proof of Proposition 1. Substitute the market-clearing conditions into the maximized objectives of the universities. Each university maximizes a continuous function on the convex set [[theta].bar], [bar.[theta]]]. Therefore, by Berge's maximum theorem, there is an upper hemicontinuous best-response relation [[theta].sup.*.sub.u](K). Labor-market clearing implies that [K.sup.*]([{[[theta].sup.*.sub.u]}.sub.u[member of]U) is a continuous function of its arguments, since the densities are positive on their support. Substituting the best responses of the two types of universities into the market-clearing condition yields a upper hemicontinuous correspondence [K.sup.*](K), defined on [[theta].bar][[mu].bar], [bar.[theta]][bar.[mu]] to itself, which has a fixed point by Kakutani's theorem.

Now suppose there were multiple equilibrium standards, [K.sub.1] and [K.sub.2], for good job assignments, with [K.sub.1] > [K.sub.2]. Then in equilibrium with the minimal productivity standard [K.sub.2], each university will place more students at good jobs than in equilibrium with [K.sub.1]--all students that had good jobs when [K.sub.1] is the standard will have a good job when [K.sub.2] is the standard. But then the labor markets cannot clear for both [K.sub.1] and [K.sub.2]. Further, facing a single equilibrium standard K, the measure of students from type u university is pinned down by optimization--each university chooses a grading standard that maximizes the expected product of those receiving good jobs.

Proof of Proposition 2. Suppose only students with "A" grades receive good jobs. Equilibrium requires EHA[[theta]] = [E.sub.IA][[theta]]. Define [Q.sub.u](x) = ([[integral].sup.[bar.[theta]].sub.x][f.sub.u](t)dt)/ ([[integral].sup.[bar.[theta]]].sub.x][f.sub.u](t)dt) for x [member of] [[[theta].bar], [[theta].bar]), with [Q.sub.u]([bar.[theta]]) = [bar.[theta]], to be the expected productivity of "A" students given any standard x; [Q.sub.U](x) is trivially strictly increasing in x. By CFOSD, [Q.sub.H](x) > [Q.sub.I](x) for all x [member of] [[[theta].bar], [bar.[theta]]), so the value y defined by [Q.sub.H](y) = [Q.sub.I](x) is less than x.

Proof of Proposition 3. Because "A" students are hired before "B" students, the social planner gives "A" to all students who, in her opinion, should be employed on a good job, and the labor market assigns good jobs only to "A" students. The first-order conditions to the social planner's problem are:

(A1) -[[??].sub.u][f.sub.u]([[??].sub.u]) + [lambda][f.sub.u]([[??].sub.u]) = 0,

where [lambda] is the Lagrange multiplier for the capacity constraint of the social planner problem. By the full support assumption, the densities are positive, so the first-order conditions simplify to [[??].sup.P.sub.u] = [lambda].

Proof of Proposition 4. Observe that [Q.sup.-1.sub.H]([[bar.[theta]] = [Q.sup.-1.sub.I]([bar.[theta]]) = [bar.[theta]], and CFOSD implies that [Q.sup.-1.sub.H]([bar.[theta]] - [epsilon]) > [Q.sup.-1.sub.I]([bar.[theta]] - [epsilon]) for any [epsilon] positive, but sufficiently small. Thus, [Q'.sub.H] ([bar.[theta]]) < [Q'.sub.I] ([bar.[theta]]). By the implicit function theorem, [[??]'.sub.H]([[??].sub.I] | [[??].sub.I] = [bar.[theta]])= ([Q'.sub.I]([bar.[theta]])/[Q'.sub.H]([bar.[theta])) > 1. Continuity of [[??]'.sub.H]([[??].sub.I]) ensures that [[??]'.sub.H]([[??].sub.I]) > 1 over some non-degenerate interval, [[[??].sub.I], [bar.[theta]]). The result follows.

Proof of Proposition 5. By Proposition 4, there is an interval [[[??].sub.I], 1] where [[??]'.sub.H]([[??].sub.I]) is at least 1, so that [[??]'.sub.H]([[??].sub.I]) = 1 when [[??].sub.I] = [[??].sub.I]. Thus, for some [[??].sub.H] = [[??].sub.H]([[??].sub.I]) and [[??].sub.I] = [[??].sub.I], [Q.sub.H]([[??].sub.H]) = [Q.sub.I]([[??].sub.I]) and [Q'.sub.H]([[??].sub.H]) = [Q'.sub.I] ([[??].sub.I]). The second derivative of [[??].sub.H]([[??].sub.I]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, [Q.sub.H]([[??].sub.H]) = [Q'.sub.I]([[??].sub.I]), [[??].sub.H]([[??].sub.I]) is concave if [Q".sub.I]([[??].sub.I]) < [Q".sub.H]([[??].sub.H]). We now solve for the shapes of derivatives of Q(x|x) when densities have linear right tails:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Observe that Q'([[??].sub.H] | [b.sub.H]) = Q'([[??].sub.I]/[b.sub.I]) implies ([a.sub.H] + [b.sub.H])/ (2[a.sub.H] + [b.sub.H] + [b.sub.H][[theta].sub.H]) = ([a.sub.I] + [b.sub.I])/(2[a.sub.I] + [b.sub.I] + [b.sub.I][[theta].sub.i]), which in turn combined with [a.sub.H][b.sub.I](1 + [[??].sub.I]) < [a.sub.I][b.sub.H](1 + [[??].sub.H]) implies ([b.sub.H])/([b.sub.H](1 + [[??].sub.H]) + 2[a.sub.H]) > ([b.sub.I]/([b.sub.I](1 + [[??].sub.I]) + 2[a.sub.H]). Multiply both sides of this inequality of derivatives by ([a.sub.H] + [b.sub.H])/([b.sub.H](1 + [[??].sub.H]) + 2[a.sub.H]) and ([b.sub.I] + [a.sub.I])/([b.sub.I] (1 + [[??].sub.I]) + 2[a.sub.I]), respectively, and multiply by 4/3:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, [[??].sub.H]([[??].sub.I]) is concave at [??]. Since ([partial derivative][[??].sub.H])/ ([partial derivative], [[??].sub.I]) > 1 for every [[??].sub.I] > [[??].sub.I], concavity of [[??].sub.H]([[??].sub.I]) at [??] contradicts [[??]'.sub.H]([[??].sub.I]) = 1.

Proof of Corollary 2. -[f.sub.u] ([[theta].sub.u])d[[??].sub.u]. is the increase in "A"s at university u responding to d[GAMMA]. As increasing F causes grading standards to fall and (d[[??].sub.H])/(d[[??].sub.I])> 1, the result follows.

Proof of Proposition 6. University u solves:

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The associated first-order conditions for interior solution are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The densities are positive everywhere. Integrating and rearranging terms yields

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solve for [[??].sub.u]:

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

R([mu]) is an increasing function since [g([mu])[[mu].sup.2]]' is positive for all Ix in the support:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equilibrium is governed by Equations (A3) and (A4). The latter is a decreasing curve in ([??], [??]) space (see Figure 3) so there is a unique equilibrium described by their intersections. If the optimal solution is on the boundary with [??] = [[mu].bar]. the optimality condition becomes R([[mu].bar]) [greater than or equal to] [??]; if it is at [??] = [[theta].bar], the optimality condition becomes R([??]) _ [less than or equal to] [[theta].bar].

The CFOSD assumption implies that [E.sub.I][[theta]|[theta] > [??]] < [E.sub.H][[theta]|[theta] > [??]], for every [??] < [bar.[theta]]. Define [Q.sub.u]([[??].sub.u]) [equivalent to] K/ ([E.sub.u][[theta]|[theta] > [[??].sub.u]); then [Q.sub.H](x) < [Q.sub.I](x), [for all]x [member of] [[[theta].bar], [theta]), and note that [Q.sub.u]([[??].sub.u]) is decreasing in [[??].sub.u]. Then [[??].sup.*.sub.uA] = [Q.sub.u] ([[??].sup.*.sub.uA])) implies

[[??].sup.*.sub.HA] = [Q.sub.H](R([[??].sup.*.sub.HA])) < [Q.sub.I](R([[??].sup.*.sub.HA])).

Since [Q.sub.I](R(x)) is decreasing, [[??].sup.*.sub.IA], a solution to [Q.sub.I](R(x)) = x, has to be to the right of [[??].sup.*.sub.HA]. Since R is increasing, [[??].sup.*.sub.H] = R([[??].sup.*.sub.HA]) < R([[??].sup.*.sub.IA]) = [[??].sup.*.sub.I].

Proof of Proposition 7. The social planner chooses grading standards to maximize expected output subject to the constraint that job placement decisions are based on student social skills and the information contained in grades:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first-order conditions (assuming an interior solution) for a type u school are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The densities cancel on both sides. After canceling and isolating [lambda] on the right-hand side, the equations for both types of universities become

(A6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(A7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Optimality condition, Equation (A7), is the same for both university types. If the optimal solution is on the boundary, [[??].sub.u] = [bar.[mu]], the condition is [[??].sub.u],E[[mu]] [greater than or equal to] [lambda]; if it is at [[??].sub.u] = [bar.[theta]], the condition becomes [bar.[theta]]E[[mu]|[mu] < [[??].sub.u]] [less than or equal to] [lambda]. Denote the implicit function for [[??].sub.u] from Equation (A6) by [Q.sub.u]([??]) and that from Equation (A7) by R([??]). Their derivatives are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because both [F.sub.u](x) and G(x) feature increasing hazards, the derivatives of both expectations are less than 1 (Bagnoli and Bergstrom 2005). As E[[mu]|[mu] > x] > x for all x < [[mu].bar], we have 0 > R'(x) > [Q'.sub.u],(x).

By CFOSD, the efficient job assignment Equation (A6) for a type H school is everywhere to the left of that for a type I school. Therefore, the intersection of Equations (A6) and (A7) in ([??], [??]) space that determine ([[??].sub.H], [[??].sub.HA]) Occur above and to the left of the intersection that determines ([[??].sub.I], [[??].sub.IA]). Thus, [[??].sup.P.sub.H] [greater than or equal to] [[??].sup.P.sub.I] and [[??].sup.P.sub.HA] [less than or equal to] [[??].sup.P.sub.IA], and [[??].sup.P.sub.HA] = [[??].sup.P.sub.IA] occurs only when lower boundary on support of [mu] binds for both types of universities, so that all "A" students get good jobs.

Proof of Proposition 8. We write down the first-order conditions of universities u and u' (see the proof of Proposition 6 for details of derivation):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [G.sub.u] > [sub.C] G(u'), (1 - [G.sub.u'](x))/([g.sub.u'](x)) < (1 - [G.sub.u](x))/ ([g.sub.u](x)) (see Shaked and Shanthikumar 2007). Let R'(x) = K [1/x - (1/[x.sup.2])(1 - [G.sup.u'](x))/([g.sup.u'](x))] and R(x) = K [1/x - (1/[x.sup.2])(1 - [G.sup.u](x))/([g.sub.u](x))]. Then R' (x) > R(x). Define Q(t) implicitly as the solution to E[[theta]|u, [theta] > Q(t)] = E[[theta]|u', [theta] > Q(t)] = K/t. Then we can rewrite these first-order conditions as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

R([bar.[mu]]) = K/[bar.[mu]] > Q([bar.[mu]]) since Q(x) < x; same holds for R'(x). The intersections of Q(x) and R(x) and of Q(x) and R'(x) are unique. This fact together with R(x)< R'(x) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that the solution to R'(x) = Q(x) lies on [[[mu].bar], [[??].sup.*.sub.u]), or [[??].sup.*.sub.u'], < [[??].sup.*.sub.u], and since Q(x) is decreasing, [[??].sup.*.sub.u] = Q([[??].sup.*.sub.u]) < Q([[??].sup.*.sub.u'])= [[??].sup.*.sub.u'].

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Bagues, M., M. S. Labini, and N. Zinovyeva. "Differential Grading Standards and University Funding: Evidence from Italy." CESifo Economic Studies, 54(2), 2008, 149-76.

Bar, T., and A. Zussman. "Partisan Grading." American Economic Journal: Applied Economics, 4(1), 2012, 30-48.

Betts, J. R. "The Impact of Educational Standards on the Level and Distribution of Earnings." The American Economic Review. 88(1), 1998, 266-75.

Boleslavsky, R., and C. Cotton. "Grade Inflation and Education Quality." Working Paper, University of Miami, 2012.

Card, D., and J. E. DiNardo. "Skill-Biased Technological Change and Rising Wage Inequality: Some Problems and Puzzles." Journal of Labor Economics, 20(4), 2002, 733-83.

Chan, W., L. Hao, and W. Suen. "'A Signaling Theory of Grade Inflation." International Economic Review, 48(3), 2007, 1065-90.

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Costrell, R. "A Simple Model of Educational Standards." American Economic Review, 84(4), 1994, 956-71.

Dubey, P., and J. Geanakoplos. "Grading Exams: 100, 99, ..., 1 Or A, B, C? Incentives in Games of Status." Games and Economic Behavior, 69(1), 2010, 72-94.

Katz, L. F., and D. H. Autor. "Changes in the Wage Structure and Earnings Inequality," in Handbook of Labor Economics, Vol. 3, Chapter 26, edited by O. C. Ashenfelter and D. E. Card. Amsterdam, The Netherlands: North-Holland, 1999, 1463-555.

Katz, L. F., and K. M. Murphy. "Changes in Relative Wages, 1963-1987: Supply and Demand Factors." The Quarterly Journal of Economics, 107(1), 1992, 35-78.

MacLeod, W. B., and M. Urquiola. "'Anti-Lemons: School Reputation and Educational Quality." Working Paper 15112, National Bureau of Economic Research, 2009.

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Popov, S. V. "Three Essays in Economics." Ph.D. Thesis, University of Illinois, 2011.

Rojstaczer, S. "Where All Grades Are Above Average." Washington Post, 2003.

Rose, H., and J. R. Betts. "The Effect of High School Courses on Earnings." Review of Economies and Statistics, 86(2), 2004, 497-513.

Shaked, M., and J. G. Shanthikumar. Stochastic Orders, New York: Springer, 2007.

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SERGEY V. POPOV and DAN BERNHARDT *

* We thank Angelo Mele, Sara LaLumia, Taras Pogorelov, Bait Taub, Grigory Kosenok, and Alexei Savvateev and participants at the 2010 Midwest Economic Association Meetings, the 2010 Missouri Economics Conference, Washington University, St. Louis, Novosibirsk State University (Russia), and Higher School of Economics (Russia) for helpful comments. Popov gratefully acknowledges the support from the Basic Research Program at HSE. All errors are ours.

Popov: Docent, National Research University, Higher School of Economics, Moscow 109028, Russia. Phone +79150897830, Fax +74956288606, E-mail svp@hse.ru

Bernhardt: IBE Professor of Economics and Finance. University of Illinois, Urbana, IL 61801. Phone +12172445708, Fax +12173330120, E-mail danber@ illinois.edu

(1.) There is a large empirical grading standard literature; see Bar and Zussman (2012), Rose and Betts (2004), and Bagues, Labini, and Zinovyeva (2008) for both the questions they study and their literature reviews.

(2.) For evidence of the increase in good jobs with complementarities between skill and job quality see Katz and Murphy (1992), Katz and Autor (1999). Card and DiNardo (2002), etc.

(3.) This is equivalent to a hazard ordering, ([f.sub.H] (x))/(1 - [F.sub.H](x)) < ([f.sub.I](x))/(1 - [F.sub.I](x))[for all]x, which is implied by likelihood ordering: ([f.sub.H](x)/[f.sub.I][(x))'.sub.x] > 0, see Shaked and Shanthikumar (2007).

(4.) Qualitatively identical outcomes emerge if firms pay students wages that equal their expected productivity given the information revealed by their university and grading standard, grade, and social skill (see Popov 2011).

(5.) Equivalently, universities could treasure academic integrity so that only [theta] affects grades.

(6.) If the denominator is 0, we set [E.sub.uB][[theta]] = [theta] and [E.sub.uA][[theta]] = [bar.[theta]] to preserve continuity; I(x) is the indicator function.

(7.) If wages were not kept fixed, but instead were adjusting to expected productivities of employed students, then universities seek to maximize the total productivity of its students at good jobs. Qualitatively identical outcomes emerge.

(8.) One example of such densities would be [f.sub.I] (x) = a - bx and [f.sub.H](x) = [lambda] [f.sub.I] (x) + (1 - [lambda])2x for [lambda] [member of] [0, 1], where a and b make [f.sub.I] a well-defined density.

TABLE 1
Evolution of GPAs at Selected Universities

 1960- 1980-
 1960 1980 2000 2000 2000

Harvard University 2.7 3.05 3.41 0.71 0.36
Princeton University 2.83 3.13 3.36 0.53 0.23
Yale University 2.56 3.27 3.48 0.92 0.21
Columbia University 3.2 3.36 0.16
Stanford University 2.98 3.27 3.55 0.57 0.28
Northwestern University 3.02 3.35 0.33
University of Chicago 2.5 3.26 0.76
MIT 2.47 3.27 3.26 0.79 -0.01
Dartmouth College 2.47 3.06 3.33 0.86 0.27
Duke University 2.41 3.02 3.36 0.95 0.34
Average grade inflation 0.7613 0.2411
University of Illinois 2.77 3.12 0.35
University of Miami 2.7 3.05 0.35
Penn State University 2.86 2.99 0.13
University of Wisconsin 2.51 2.89 3.13 0.62 0.24
University of Texas 2.6 3 0.4
University of Washington 2.31 2.97 3.12 0.81 0.15
University of California, 2.9 2.95 0.05
 Irvine
Lehigh University 2.6 2.97 0.37
Georgia Tech 2.78 2.97 0.19
George Washington University 3.03 3.25 0.22
Average grade inflation 0.5933 0.2563

GPA data from GradeInflation.com. When year X was not available,
the closest year was used; if there was no observation in [+ or -]
9 years, the observation was treated as missing. The top list of
schools is from the top list of US News National Universities
rankings, and the bottom list is from the bottom 50 and up of the
same list for the sub-samples that had grade points in at least two
time periods. Grade inflation is calculated as difference in GPAs.